LO 1.3: Estimate risk measures by estimating quantiles.

LO 1.3: Estimate risk measures by estimating quantiles.
A more general risk measure than either VaR or ES is known as a coherent risk measure. A coherent risk measure is a weighted average of the quantiles of the loss distribution where the weights are user-specific based on individual risk aversion. ES (as well as VaR) is a special case of a coherent risk measure. When modeling the ES case, the weighting function is set to [1 / (1 confidence level)] for all tail losses. All other quantiles will have a weight of zero.
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2018 Kaplan, Inc.
Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3
Under expected shortfall estimation, the tail region is divided into equal probability slices and then multiplied by the corresponding quantiles. Under the more general coherent risk measure, the entire distribution is divided into equal probability slices weighted by the more general risk aversion (weighting) function.
This procedure is illustrated for n = 10. First, the entire return distribution is divided into nine (i.e., n 1) equal probability mass slices at 10%, 20%, …, 90% (i.e., loss quantiles). Each breakpoint corresponds to a different quantile. For example, the 10% quantile (confidence level = 10%) relates to 1.2816, the 20% quantile (confidence level = 20%) relates to 0.8416, and the 90% quantile (confidence level = 90%) relates to 1.2816. Next, each quantile is weighted by the specific risk aversion function and then averaged to arrive at the value of the coherent risk measure.
This coherent risk measure is more sensitive to the choice of n than expected shortfall, but will converge to the risk measures true value for a sufficiently large number of observations. The intuition is that as n increases, the quantiles will be further into the tails where more extreme values of the distribution are located.